How to get all of the meshes that are attached to an armature via Python?
You can loop through all objects and check if they are parented to the armature.

rig = bpy.data.objects['rig']
for obj in bpy.data.objects:
    if obj.parent == rig:
        # Do stuff
        pass


If some objects are only connected via the armature modifier, simply check that the rig is the object of the armature modifier. 

rig = bpy.data.objects['rig']
for obj in bpy.data.objects:
    if (obj.type == 'MESH' and
        rig in [m.object for m in obj.modifiers if modifier.type == 'ARMATURE']
        ):
        # Do stuff
        pass